There is not enough information here. Are you going to be putting a walkway?

I'm hoping to bump this up as I want to know the answer, too. The Bermuda in my garden tends not to grow rhizomes below 12", so that would be, in my mind, a minimum depth for a root barrier. 18" to be super safe, as I've dug out plenty of rhizomes at the 10-12" line. I would like to know other's thoughts on thickness.

Sorry for the delay. No walkway....my neighbor's lawn is weeds and bermuda and it has creeped into my garden beds and St. Augustine. I have sprayed but it seems like a losing battle. So I am thinking of a physical barrier.

Sadly, Bermuda grass spread by above ground stolons, underground rhizomes and by seed. Blocking the roots won't be enough.

Root barrier will sure help. May be it will not solve the situation entirely, but will greatly help. You should put it from about 2 cm above the ground level down. Sorry, I can't advise you on the thickness. After installing you will have to remove the grass roots entirely from your side of the barrier, and watch for any signs of the grass later.